Computational Topology: An Introduction (Google eBook)

Front Cover
American Mathematical Soc., 2010 - Mathematics - 241 pages
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Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.
  

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Contents

Graphs
3
12 Curves in the Plane
9
13 Knots and Links
13
14 Planar Graphs
18
Exercises
24
Surfaces
27
II2 Searching a Triangulation
33
II3 Selfintersections
37
Exercises
123
Morse Functions
125
VI2 Transversality
130
VI3 Piecewise Linear Functions
135
VI4 Reeb Graphs
140
Exercises
145
Computational Persistent Topology
147
Persistence
149

II4 Surface Simplification
42
Exercises
47
Complexes
51
III2 Convex Set Systems
57
III3 Delaunay Complexes
63
III4 Alpha Complexes
68
Exercises
74
Computational Algebraic Topology
77
Homology
79
IV2 Matrix Reduction
85
IV3 Relative Homology
90
IV4 Exact Sequences
95
Exercises
101
Duality
103
V2 Poincaré Duality
108
V3 Intersection Theory
114
V4 Alexander Duality
118
VII2 Efficient Implementations
156
VII3 Extended Persistence
161
VII4 Spectral Sequences
166
Exercises
171
Stability
175
VIII2 Stability Theorems
180
VIII3 Length of a Curve
185
VIII4 Bipartite Graph Matching
191
Exercises
197
Applications
199
IX2 Elevation for Protein Docking
206
IX3 Persistence for Image Segmentation
213
IX4 Homology for Root Architectures
218
Exercises
224
References
227
Index
235
Copyright

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About the author (2010)

Herbert Edelsbrunner is Arts and Sciences Professor of Computer Science at Duke University. He was the winner of the 1991 Waterman award from the National Science Foundation and is the founder and director of Raindrop Geomagic, a 3-D modelling company.

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